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Theorem abeq1 2908
Description: Equality of a class variable and a class abstraction. Commuted form of abeq2 2907. (Contributed by NM, 20-Aug-1993.)
Assertion
Ref Expression
abeq1 ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abeq1
StepHypRef Expression
1 abeq2 2907 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 eqcom 2804 . 2 ({𝑥𝜑} = 𝐴𝐴 = {𝑥𝜑})
3 bicom 214 . . 3 ((𝜑𝑥𝐴) ↔ (𝑥𝐴𝜑))
43albii 1915 . 2 (∀𝑥(𝜑𝑥𝐴) ↔ ∀𝑥(𝑥𝐴𝜑))
51, 2, 43bitr4i 295 1 ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wal 1651   = wceq 1653  wcel 2157  {cab 2783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793
This theorem is referenced by:  rabeqc  3552  disj  4210  euabsn2  4447  dm0rn0  5543  dffo3  6598  dfsup2  8590  rankf  8905  dfon3  32503  dfiota3  32534  dffo3f  40105
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